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Archive for the ‘module theory’ Category

Let A be an abelian group and T = \{ T_i : A \to A \} be a collection of endomorphisms of A. The commutant T' of T is the set of all endomorphisms of A commuting with every element of T; symbolically,

\displaystyle T' = \{ S \in \text{End}(A) : TS = ST \}.

The commutant of T is equal to the commutant of the subring of \text{End}(A) generated by the T_i, so we may assume without loss of generality that T is already such a subring. In that case, T' is just the ring of endomorphisms of A as a left T-module. The use of the term commutant instead can be thought of as emphasizing the role of A and de-emphasizing the role of T.

The assignment T \mapsto T' is a contravariant Galois connection on the lattice of subsets of \text{End}(A), so the double commutant T \mapsto T'' may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

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The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even \mathbb{Z} fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one with the property that every element r \in R acts nontrivially in some simple (left) module M.

Said another way, let the Jacobson radical J(R) of a ring consist of all elements of r which act trivially on every simple module. By definition, this is an intersection of kernels of ring homomorphisms, hence a two-sided ideal. A ring R is then semiprimitive if it has trivial Jacobson radical.

The goal of this post will be to discuss some basic properties of the Jacobson radical. I am again working mostly from Lam’s A first course in noncommutative rings.

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In the previous post we learned that it is possible to recover the center Z(R) of a ring R from its category R\text{-Mod} of left modules (as an \text{Ab}-enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings R, S are said to be Morita equivalent if the categories R\text{-Mod}, S\text{-Mod} are equivalent as \text{Ab}-enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the bicategory of bimodules, which we briefly describe.

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The center Z(G) of a group is an interesting construction: it associates to every group G an abelian group Z(G) in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor \text{Grp} \to \text{Ab} (unlike the abelianization G/[G, G]). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category C we may associate to an object c \in C its automorphism group \text{Aut}(c) or endomorphism monoid \text{End}(c)), and this is a canonical construction which again doesn’t extend in an obvious way to a functor C \to \text{Grp} or C \to \text{Mon}. (It merely reflects some special part of the bifunctor \text{Hom}(-, -).)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group G, but of the identity functor G \to G, where G is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.

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In this post, I’d like to record a few basic definitions and results regarding noncommutative rings. This is a subject clearly of great importance and generality, but I haven’t had much exposure to it, and I’m trying to fix that. I am working mostly from Lam’s A first course in noncommutative rings.

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(Commutative) Poisson algebras are clearly very interesting, so it would be nice to have ways of constructing examples. We know that k[x, p] is a Poisson algebra with bracket uniquely defined by \{ x, p \} = 1; this describes a classical particle in one dimension, and is the classical limit of a quantum particle in one dimension (essentially the Weyl algebra).

More generally, if A, B are Poisson algebras, then the tensor product A \otimes_k B can be given a Poisson bracket given by extending

\displaystyle \{ a_1 \otimes b_1, a_2 \otimes b_2 \} = \{ a_1, a_2 \} \otimes b_1 b_2 + a_2 a_1 \otimes \{ b_1, b_2 \}

linearly. At least when A, B are unital, this Poisson algebra is the universal Poisson algebra with Poisson maps from A, B such that the images of elements of A Poisson-commute with the images of elements of B. In particular, it follows that k[x_1, p_1, ..., x_n, p_n] is a Poisson algebra with the bracket

\{ x_i, x_j \} = \{ p_i, p_j \} = 0, \{ x_i, p_j \} = \delta_{ij}.

This describes a classical particle in n dimensions, or n different classical particles in one dimension, and it is the classical limit of a quantum particle in n dimensions, or n different quantum particles in one dimension.

Today we’ll discuss the question of how one might go about constructing Poisson brackets more generally.

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Charles Siegel over at Rigorous Trivialities suggested a NaNoWriMo for math bloggers: instead of writing a 50,000-word novel, just write a blog post every day. I have to admit I rather like the idea, so we’ll see if I can keep it up.

Continuing the previous post, what we want to do now is to think of restriction \text{Res}_H^G :  \text{Rep}(G) \to \text{Rep}(H) as a forgetful functor, since restricting a representation just corresponds to forgetting some of the data that defines it. Its left adjoint, if it exists, should be a construction of the “free G-representation” associated to an H-representation. Given a representation \rho : H \to \text{Aut}(V) we therefore want to find a representation \rho' : G \to \text{Aut}(V') with the following universal property: any H-intertwining operator \phi : V \to W for \tau a G-representation on W naturally determines a unique G-intertwining operator \phi' : V' \to W. In other words, we want to construct a functor \text{Ind}_G^H : \text{Rep}(H) \to \text{Rep}(G) such that

\text{Hom}_{\text{Rep}(G)}(\text{Ind}_H^G \rho, \tau) \simeq \text{Hom}_{\text{Rep}(H)}(\rho, \text{Res}_H^G \tau).

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